Polynomial formulas for higher genus
were first obtained by Getzler for \(g=2\)  and \(n \le 3\)  in [1]}
and then by Bergström for \(g=2\)  and \(g=3\) 
for some \(n\)  in [2]}, [3]}. Since these tell us about the cohomology, which is a representation space of \(\mathfrak {S}_n\) , the answer is
phrased in terms of Schur functions \({\bf s}_{\lambda }\) , where
\(\lambda \)  runs through the partitions of \(n\)  and these \(\lambda \) 
correspond to the irreducible representations of \(\mathfrak {S}_n\) .
Recall that the Schur functions form a basis for the symmetric functions, like the elementary or complete symmetric functions. We
refer for the representation theory to [4]}.
We write the answer as
\(\sum _{\lambda \vdash n} P_{\lambda }(q) \, {\bf s}_{\lambda }\quad \text{\rm with} \quad P_{\lambda }(q)= \frac{1}{n!} \sum _{\sigma \in \mathfrak {S}_n}\chi _{\lambda }(\sigma ) \, | {\mathcal {M}}_{g,n}^{F\cdot \sigma }| \, ,\) 
